Question

Using De Moivre’s theorem, find the least positive integer n such that `((2i)/(1+i))^n`  is a positive integer.

Answer 1

We have,  `(2i)/(1+i) = (2i)/(1+i) xx (1-i)/(1-i)`

= `(2(1+i))/(2) = 1 + i`

Let  1 + i = r cos θ + i r sin θ 
⇒ r cos θ  = 1, r sin θ  = 1
∴ r² (cos² θ  + sin² θ ) = (1)² + (1)² 
   r² = 2 ⇒ r = `sqrt2`

and  tan θ = `(1)/(1)`

tan θ  = tan `(π/4)`

θ = `(π)/(4)`

`((2i)/(1+i)) = sqrt2 ( cos  (π)/(4) + i sin  (π)/(4))`

`((2i)/(1+i))^n = [ sqrt2 ( cos  (π)/(4) + i sin  (π)/(4)]^n`

= `2^(n/2) ( cos  (nπ)/(4) + i sin  (nπ)/(4))`

Which is a positive integer

If  `(nπ)/(4)` = 0, 2π, 4π, 6π, ...

⇒ n = 0, 8, 16, 24, ...

⇒ The least value of n is 4